Consider a desired target that is buried in both interference and noise. A transmit signal excites both the desired target and the interference simultaneously. The interference and/or interferences can be foliage returns in the form of clutter for radar, scattered returns of the transmit signal from the sea-bottom and different ocean-layers in the case of sonar, or multipath returns in a communication scene. The interference returns can also include jamming signals. In all these cases, like the target return, the interference returns are also transmit signal dependent, and hence it puts conflicting demands on the receiver. In general, the receiver input is comprised of target returns, interferences and the ever present noise. The goal of the receiver is to enhance the target returns and simultaneously suppress both the interference and noise signals. In a detection environment, a decision regarding the presence or absence of a target is made at some specified instant t=to using output data from a receiver, and hence to maximize detection, the Signal power to average Interference plus Noise Ratio (SINR) at the receiver output can be used as an optimization goal. This scheme is illustrated in FIG. 1.
The transmitter output bandwidth can be controlled using a known transmitter output filter having a transfer function P1(ω) (see FIG. 2B). A similar filter with transform characteristics P2(ω) can be used at a receiver input 22a shown in FIG. 1, to control the receiver processing bandwidth as well.
The transmit waveform set f(t) at an output 10a of FIG. 1, can have spatial and temporal components to it, each designated for a specific goal. A simple situation is that shown in FIG. 2A where a finite duration waveform f(t) of energy E is to be designed. Thus the total energy can be expressed in the time domain or frequency domain as
                                          ∫            0                          T              o                                ⁢                                                                                      f                  ⁡                                      (                    t                    )                                                                              2                        ⁢                          ⅆ              t                                      =                                            1                              2                ⁢                π                                      ⁢                                          ∫                                  -                  ∞                                                  +                  ∞                                            ⁢                                                                                                              F                      ⁡                                              (                        ω                        )                                                                                                  2                                ⁢                                  ⅆ                  ω                                                              =                      E            .                                                  ⁢            Here                                              (        1        )                                          F          ⁡                      (            ω            )                          =                              ∫                          -              ∞                                      +              ∞                                ⁢                                    f              ⁡                              (                t                )                                      ⁢                          ⅇ                                                -                  jω                                ⁢                                                                  ⁢                t                                      ⁢                          ⅆ              t                                                          (        2        )            refers to the Fourier transform of the transmit waveform f(t).
Usually, transmitter output filter 12 characteristics P1(ω), such as shown in FIG. 2B, are known and for design purposes, it can be incorporated into the target transform and clutter spectral characteristics. Similarly, the receiver input filter if any (which may be at the input to the receiver 22) can be incorporated into the target transform as well as the clutter and noise spectra. Here onwards we will assume such to be the case.
Let q(t)⇄Q(ω) represent the target impulse response and its transform. In general q(t) can be any arbitrary waveform. Thus the modified target that accounts for the transmitter output filter has transform P1(ω)Q(ω). Here onwards, we shall refer to this modified form as the “target transform”, and the associated inverse transform as the “target” response signal and represent them simply by Q(ω) and q(t) respectively. In a linear domain setup, the transmit signal f(t) interacts with the target q(t), or target 14 shown in FIG. 1, to generate the output below (referred to in S. U. Pillai, H. S. Oh, D. C. Youla, and J. R. Guerci, “Optimum Transmit-Receiver Design in the Presence of Signal-Dependent Interference and Channel Noise”, IEEE Transactions on Information Theory, Vol. 46, No. 2, pp. 577-584, March 2000 and S. M. Kay, J. H. Thanos, “Optimal Transmit Signal Design for Active Sonar/Radar”, Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, 2002 (ICASSP 02), Vol. 2, pp. 1513-1516, 2002):
                              s          ⁡                      (            t            )                          =                                            f              ⁡                              (                t                )                                      *                          q              ⁡                              (                t                )                                              =                                    ∫              0                              T                o                                      ⁢                                          f                ⁡                                  (                  τ                  )                                            ⁢                              q                ⁡                                  (                                      t                    -                    τ                                    )                                            ⁢                              ⅆ                τ                                                                        (        3        )            that represents the desired signal.
The interference returns are usually due to the random scattered returns of the transmit signal from the environment, and hence can be modeled as a stochastic signal wc(t) that is excited by the transmit signal f(t). If the environment returns are stationary, then the interference can be represented by the interference power spectrum Gc(ω). Once again, if a transmit output filter is present then |P1(ω)|2Gc(ω) represents the modified interference spectrum. Here onwards, the interference power spectrum so modified will be referred simply by Gc(ω). This gives the average interference power at the receiver input to be Gc(ω)|F(ω)|2 . Finally let n(t) represent the receiver 22 input noise with power spectral density Gn(ω). Thus the receiver input signal at input 22a equalsr(t)=s(t)+wc(t)*f(t)+n(t),   (4)and the receiver input interference plus noise power spectrum equalsGI(ω)=Gc(ω)|F(ω)|2+Gn(ω).   (5)The received signal is presented to the receiver 22 at input 22a with impulse response h(t). The general receiver is of the noncausal type.
With no restrictions on the receiver 22 of FIG. 1 such as causal or noncausal, its output signal component and interference/noise components at output 22b in FIG. 1 are given by
                                                        y              s                        ⁡                          (              t              )                                =                                                    s                ⁡                                  (                  t                  )                                            *                              h                ⁡                                  (                  t                  )                                                      =                                          1                                  2                  ⁢                  π                                            ⁢                                                ∫                                      -                    ∞                                                        +                    ∞                                                  ⁢                                                      S                    ⁡                                          (                      ω                      )                                                        ⁢                                      H                    ⁡                                          (                      ω                      )                                                        ⁢                                      ⅇ                                          jω                      ⁢                                                                                          ⁢                      t                                                        ⁢                                      ⅆ                    ω                                                                                      ⁢                                  ⁢        and                            (        6        )                                                      y            n                    ⁡                      (            t            )                          =                              {                                                                                w                    c                                    ⁡                                      (                    t                    )                                                  *                                  f                  ⁡                                      (                    t                    )                                                              +                              n                ⁡                                  (                  t                  )                                                      }                    *                                    h              ⁡                              (                t                )                                      .                                              (        7        )            The output yn(t) represents a second order stationary stochastic process with power spectrum below (referred to in the previous publications and in Athanasios Papoulis, S. Unnikrishna Pillai, Probability, Random Variables and Stochastic Processes, McGraw-Hill Higher Education, New York 2002):Go(ω)=(Gc(ω)|F(ω)|2+Gn(ω)|H(ω)|2   (8)and hence the total output interference plus noise power is given by
                                                                        σ                                  I                  +                  N                                2                            =                                                1                                      2                    ⁢                    π                                                  ⁢                                                      ∫                                          -                      ∞                                                              +                      ∞                                                        ⁢                                                                                    G                        o                                            ⁡                                              (                        ω                        )                                                              ⁢                                          ⅆ                      ω                                                                                                                                              =                                                1                                      2                    ⁢                    π                                                  ⁢                                                      ∫                                          -                      ∞                                                              +                      ∞                                                        ⁢                                                            (                                                                                                                                  G                              c                                                        ⁡                                                          (                              ω                              )                                                                                ⁢                                                                                                                                                  F                                ⁡                                                                  (                                  ω                                  )                                                                                                                                                    2                                                                          +                                                                              G                            n                                                    ⁡                                                      (                            ω                            )                                                                                              )                                        ⁢                                                                                                                    H                          ⁡                                                      (                            ω                            )                                                                                                                      2                                        ⁢                                                                  ⅆ                        ω                                            .                                                                                                                              (        9        )            Referring back to FIG. 1, the signal component ys(t) in equation (6) at the receiver output 22b needs to be maximized at the decision instant to in presence of the above interference and noise. Hence the instantaneous output signal power at t=to is given by the formula below shown in S. U. Pillai, H. S. Oh, D. C. Youla, and J. R. Guerci, “Optimum Transmit-Receiver Design in the Presence of Signal-Dependent Interference and Channel Noise”, IEEE Transactions on Information Theory, Vol. 46, No. 2, pp. 577-584, March 2000, which is incorporated by reference herein:
                              P          o                =                                                                                            y                  s                                ⁡                                  (                                      t                    o                                    )                                                                    2                    =                                                                                                          1                                          2                      ⁢                      π                                                        ⁢                                                            ∫                                              -                        ∞                                                                    +                        ∞                                                              ⁢                                                                  S                        ⁡                                                  (                          ω                          )                                                                    ⁢                                              H                        ⁡                                                  (                          ω                          )                                                                    ⁢                                              ⅇ                                                  jω                          ⁢                                                                                                          ⁢                                                      t                            o                                                                                              ⁢                                              ⅆ                        ω                                                                                                                        2                        .                                              (        10        )            This gives the receiver output SINR at t=to to be the following as specified in Pillai et. al., “Optimum Transmit-Receiver Design in the Presence of Signal-Dependent Interference and Channel Noise”, incorporated herein by reference:
                    SINR        =                                            P              o                                      σ                              1                +                N                            2                                =                                                                                                                                  1                                              2                        ⁢                        π                                                              ⁢                                                                  ∫                                                  -                          ∞                                                                          +                          ∞                                                                    ⁢                                                                        S                          ⁡                                                      (                            ω                            )                                                                          ⁢                                                  H                          ⁡                                                      (                            ω                            )                                                                          ⁢                                                  ⅇ                                                      jω                            ⁢                                                                                                                  ⁢                                                          t                              o                                                                                                      ⁢                                                  ⅆ                          ω                                                                                                                                      2                                                              1                                      2                    ⁢                    π                                                  ⁢                                                      ∫                                          -                      ∞                                                              +                      ∞                                                        ⁢                                                                                    G                        I                                            ⁡                                              (                        ω                        )                                                              ⁢                                                                                                                    H                          ⁡                                                      (                            ω                            )                                                                                                                      2                                        ⁢                                          ⅆ                      ω                                                                                            .                                              (        11        )            We can apply Cauchy-Schwarz inequality in equation (11) to eliminate H(ω). This gives
                              SINR          ≤                                    1                              2                ⁢                π                                      ⁢                                          ∫                                  -                  ∞                                                  +                  ∞                                            ⁢                                                                                                                                        S                        ⁡                                                  (                          ω                          )                                                                                                            2                                                                              G                      I                                        ⁡                                          (                      ω                      )                                                                      ⁢                                  ⅆ                  ω                                                                    =                                            1                              2                ⁢                π                                      ⁢                                          ∫                                  -                  ∞                                                  +                  ∞                                            ⁢                                                                                                                                                                    Q                          ⁡                                                      (                            ω                            )                                                                                                                      2                                        ⁢                                                                                                                    F                          ⁡                                                      (                            ω                            )                                                                                                                      2                                                                                                                                                    G                          c                                                ⁡                                                  (                          ω                          )                                                                    ⁢                                                                                                                              F                            ⁡                                                          (                              ω                              )                                                                                                                                2                                                              +                                                                  G                        n                                            ⁡                                              (                        ω                        )                                                                                            ⁢                                  ⅆ                  ω                                                              =                                    SINR              max                        .                                              (        12        )            Thus the maximum obtainable SINR is given by the right side of equation (12), and this SINR is realized by the receiver design if and only if the following receiver transform referred to in previous prior art publications, is true:
                                                                                          H                  opt                                ⁡                                  (                  ω                  )                                            =                                                                                          S                      *                                        ⁡                                          (                      ω                      )                                                                                                                                                    G                          c                                                ⁡                                                  (                          ω                          )                                                                    ⁢                                                                                                                              F                            ⁡                                                          (                              ω                              )                                                                                                                                2                                                              +                                                                  G                        n                                            ⁡                                              (                        ω                        )                                                                                            ⁢                                  ⅇ                                                            -                      jω                                        ⁢                                                                                  ⁢                                          t                      o                                                                                                                                              =                                                                                                                  Q                        *                                            ⁡                                              (                        ω                        )                                                              ⁢                                                                  F                        *                                            ⁡                                              (                        ω                        )                                                                                                                                                                          G                          c                                                ⁡                                                  (                          ω                          )                                                                    ⁢                                                                                                                              F                            ⁡                                                          (                              ω                              )                                                                                                                                2                                                              +                                                                  G                        n                                            ⁡                                              (                        ω                        )                                                                                            ⁢                                                      ⅇ                                                                  -                        jω                                            ⁢                                                                                          ⁢                                              t                        o                                                                              .                                                                                        (        13        )            In equation (13), the phase shift e−jωto can be retained to approximate causality for the receiver waveform. Interestingly even with a point target (Q(ω)≡1), flat noise (Gn(ω)=σn2), and flat clutter (Gc(ω)=σc2), the optimum receiver is not conjugate-matched to the transmit signal transform F(ω) as in the classical matched filter receiver.Prior Art Transmitter Waveform Design
When the receiver design satisfies equation (13), the output SINR is given by the right side of the equation (12), where the free parameter |F(ω)|2 can be chosen to further maximize the output SINR, subject to the transmit energy constraint in equation (1). Thus the transmit signal design reduces to the following optimization problem: Maximize
                                          SINR            max                    =                                    1                              2                ⁢                π                                      ⁢                                          ∫                                  -                  ∞                                                  +                  ∞                                            ⁢                                                                                                                                                                    Q                          ⁡                                                      (                            ω                            )                                                                                                                      2                                        ⁢                                                                                                                    F                          ⁡                                                      (                            ω                            )                                                                                                                      2                                                                                                                                                    G                          c                                                ⁡                                                  (                          ω                          )                                                                    ⁢                                                                                                                              F                            ⁡                                                          (                              ω                              )                                                                                                                                2                                                              +                                                                  G                        n                                            ⁡                                              (                        ω                        )                                                                                            ⁢                                  ⅆ                  ω                                                                    ,                            (        14        )            subject to the energy constraint
                                          ∫            0                          T              o                                ⁢                                                                                      f                  ⁡                                      (                    t                    )                                                                              2                        ⁢                          ⅆ              t                                      =                                            1                              2                ⁢                π                                      ⁢                                          ∫                                  -                  ∞                                                  +                  ∞                                            ⁢                                                                                                              F                      ⁡                                              (                        ω                        )                                                                                                  2                                ⁢                                  ⅆ                  ω                                                              =                      E            .                                              (        15        )            To solve this new constrained optimization problem, combine equations (14)-(15) to define the modified Lagrange optimization function (referred to in T. Kooij, “Optimum Signal in Noise and Reverberation”, Proceeding of the NATO Advanced Study Institute on Signal Processing with Emphasis on Underwater Acoustics, Vol. I, Enschede, The Netherlands, 1968.)
                              Λ          =                                    ∫                              -                ∞                                            +                ∞                                      ⁢                                          {                                                                                                                                                                                    Q                            ⁡                                                          (                              ω                              )                                                                                                                                2                                            ⁢                                                                        y                          2                                                ⁡                                                  (                          ω                          )                                                                                                                                                                                          G                            c                                                    ⁡                                                      (                            ω                            )                                                                          ⁢                                                                              y                            2                                                    ⁡                                                      (                            ω                            )                                                                                              +                                                                        G                          n                                                ⁡                                                  (                          ω                          )                                                                                                      -                                                            1                                              λ                        2                                                              ⁢                                                                  y                        2                                            ⁡                                              (                        ω                        )                                                                                            }                            ⁢                              ⅆ                ω                                                    ⁢                                  ⁢        where                            (        16        )                                          y          ⁡                      (            ω            )                          =                                        F            ⁡                          (              ω              )                                                                    (        17        )            is the free design parameter. From equations (16)-(17),
            ∂      Λ              ∂      y        =  0gives (details omitted)
                                          ∂                          Λ              ⁡                              (                ω                )                                                          ∂            y                          =                              2            ⁢                          y              ⁡                              (                ω                )                                      ⁢                          {                                                                                                                  G                        n                                            ⁡                                              (                        ω                        )                                                              ⁢                                                                                                                    Q                          ⁡                                                      (                            ω                            )                                                                                                                      2                                                                                                  {                                                                                                                                  G                              c                                                        ⁡                                                          (                              ω                              )                                                                                ⁢                                                                                    y                              2                                                        ⁡                                                          (                              ω                              )                                                                                                      +                                                                              G                            n                                                    ⁡                                                      (                            ω                            )                                                                                              }                                        2                                                  -                                  1                                      λ                    2                                                              }                                =          0.                                    (        18        )            where Λ(ω) represents the quantity within the integral in equation (16). From equation (18), either
                                          y            ⁡                          (              ω              )                                =          0                ⁢                                  ⁢        or                            (        19        )                                                                                                                          G                    n                                    ⁡                                      (                    ω                    )                                                  ⁢                                                                                                Q                      ⁡                                              (                        ω                        )                                                                                                  2                                                                              {                                                                                    G                        c                                            ⁢                                                                                          ⁢                                              (                        ω                        )                                            ⁢                                                                        y                          2                                                ⁡                                                  (                          ω                          )                                                                                      +                                                                  G                        n                                            ⁡                                              (                        ω                        )                                                                              }                                2                                      -                          1                              λ                2                                              =          0                ,                                  ⁢                  which          ⁢                                          ⁢          gives                                    (        20        )                                                      y            2                    ⁡                      (            ω            )                          =                                                                          F                ⁡                                  (                  ω                  )                                                                    2                    =                                                                                          G                    n                                    ⁡                                      (                    ω                    )                                                              ⁢                              (                                                      λ                    ⁢                                                                                        Q                        ⁡                                                  (                          ω                          )                                                                                                                            -                                                                                    G                        n                                            ⁡                                              (                        ω                        )                                                                                            )                                                                    G                c                            ⁡                              (                ω                )                                                                        (        21        )            provided y2(ω)>0. See T. Kooij cited above incorporated by reference herein.
However, this particular method is not relevant to the current invention, since as disclosed in the next section, the current invention focuses on compressing and stretching out the total interference plus noise spectrum to be white (flat) over the desired frequency band of interest by redesigning the transmit signal transform accordingly.